The functionαα\alpha is called the (left) action of GGG on XXX. When there is no confusion, α⁢(g,x)αgx\alpha(g,x) is simply written g⁢xgxgx, so that the two conditions above read 1⁢x=x1xx1x=x and (g1⁢g2)⁢x=g1⁢(g2⁢x)subscriptg1subscriptg2xsubscriptg1subscriptg2x(g_{1}g_{2})x=g_{1}(g_{2}x).

If a topological transformation group GGG on XXX is effective, then GGG can be viewed as a group of homeomorphisms on XXX: simply define hg:X→Xnormal-:subscripthgnormal-→XXh_{g}:X\to X by hg⁢(x)=g⁢xsubscripthgxgxh_{g}(x)=gx for each g∈GgGg\in G so that hgsubscripthgh_{g} is the identity function precisely when g=1g1g=1.

If GGG is a topological group, GGG can be considered a topological transformation group on itself. There are many continuous actions that can be defined on GGG. For example, α:G×G→Gnormal-:αnormal-→GGG\alpha:G\times G\to G given by α⁢(g,x)=g⁢xαgxgx\alpha(g,x)=gx is one such action. It is continuous, and satisfies the two action axioms. GGG is also effective with respect to αα\alpha.